| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s − 2·11-s − 4·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 2·20-s − 2·22-s − 25-s − 4·26-s + 28-s + 6·29-s + 6·31-s + 32-s − 6·34-s − 2·35-s − 2·37-s − 4·38-s − 2·40-s + 2·41-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.603·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 0.426·22-s − 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s − 0.316·40-s + 0.312·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48502000245844, −13.90754983333223, −13.49061714062729, −12.83470858091550, −12.51917993340314, −11.88841477945236, −11.60456041119468, −10.98799065239162, −10.56039648585524, −10.04473434474961, −9.372646291947116, −8.597114129597858, −8.218495225721438, −7.739028654448851, −7.135496232741475, −6.605737722890841, −6.187021955090182, −5.168013331245348, −4.944007206671428, −4.264693961820678, −3.976156037144861, −3.064158860199609, −2.368255936460059, −2.103421729274806, −0.8238473372067672, 0,
0.8238473372067672, 2.103421729274806, 2.368255936460059, 3.064158860199609, 3.976156037144861, 4.264693961820678, 4.944007206671428, 5.168013331245348, 6.187021955090182, 6.605737722890841, 7.135496232741475, 7.739028654448851, 8.218495225721438, 8.597114129597858, 9.372646291947116, 10.04473434474961, 10.56039648585524, 10.98799065239162, 11.60456041119468, 11.88841477945236, 12.51917993340314, 12.83470858091550, 13.49061714062729, 13.90754983333223, 14.48502000245844
